\chapter{Simple SNIa Model}
\label{chap:Ia}

Type Ia supernovae occur in binary systems in which one of the stars is a
white dwarf while the other can be a giant star or another white dwarf.
We just take the one white dwarf case and do not consider double degenerate
case (merger of two white dwarfs) 
which is rare. And since our focus is on the nucleosynthetic part, the 
detailed composition and mechanism might not affect the final result much.
Thus we just make the white dwarf consisting of 50% Carbon-12 and 50%
Oxygen-16 and apply a simple set of hydrodynamic equations to describe it
as shown in ref{tyu:master}.

A white dwarf is the remnant of a star that has completed its normal life
cycle, say, after a red giant star phase for example. It is composed mostly
of electron-degenerate matter and the pressure inside mainly depends on the 
density but not temperature as in regular stars. If the companion of a white
dwarf is a giant star, the white dwarf may gradually accretes mass from it.
Once the white dwarf has accreted enough matter the degenerate pressure
can no longer support the star. The mass limit is called Chandrasekha limit,
which is about 1.44 solar masses. Then it will contract and release the 
gravitational potential energy which would heat up the whole star, since
the temperature grediant is small in a white dwarf. Once the
temperature passes the carbon burning threshold, the whole star will
encounter a thermonuclear blast and nothing leaves behind. In our simple
model, we just give the white dwarf a big initial mass of 1.5 solar masses
and a high initial temperature of 1.5 billion Kelvin to let it explode. 

In this chapter I will show a variety of results from our simple model.

%===============================================================================
\section{White Dwarf Equations}
%===============================================================================

The simple SNIa model is a 1-zone, 0-dimension model. We let the white dwarf 
(to explode) be a uniform, isotropic sphere with radius $R$. Thus the 
temperature and density do not depend on the radius of the white dwarf.

I am going to list the hydrodynamic equations below, which describe the white
dwarf. The detailed derivations are in my Master's thesis (ref here).
%
\begin{equation} \label{eq:dxdt}
\frac{dx}{dt} = y
\end{equation}
%
\begin{equation} \label{eq:dydt}
\frac{dy}{dt} = \frac{P_0}{\rho_0 R_0^2} \left[ \frac{P}{P_0}x^2 - \frac 1{x^2}
  \right]
\end{equation}
where $x\equiv \frac{R}{R_0}$, $y\equiv \frac{v}{R_0}$ and $R_0$ is the 
initial radius of the white dwarf.

For thermodynamic part, we change to entropy approach instead of temperature
in my master's work:
%
\begin{equation} \label{eq:dsdt}
\frac{d(s/k_B)}{dt} = \frac{dq}{k_BT} -
  \sum_i \frac{\mu_i}{k_BT} \frac{dY_i}{dt}
\end{equation}

This is the first order ordinary differential equation for temperature,
together with eq(\ref{eq:dxdt}) and eq(\ref{eq:dydt}) forming a full 
series of equations that determine the behavior of the white dwarf. 

There are three terms in the bracket on the rhs of eq(\ref{eq:dsdt}) that
determine how temperature changes. The first term is the heat loss mostly by 
neutrinos which we do not take into account in this simple model. The second
term is basically the pressure work (e.g. for classical particles 
$T\frac{\partial P}{\partial T} = P$). The third term is the energy generation
part, which will be discussed in detail later in \S \ref{sec:entropy_gen}.

%===============================================================================
\section{Entropy Generation} \label{sec:entropy_gen}
%===============================================================================

Here consider the entropy generation in each reaction and the neutrino and
photon entropy loss.

%===============================================================================
\section{Explosion and Nucleosynthesis} \label{sec:nucleosynthesis}
%===============================================================================

To solve the three first order ordinary differential equations (\ref{eq:dxdt}, 
\ref{eq:dydt}, \ref{eq:dsdt}), which determine the evolution of size, velocity
and temperature, we used {\bf gsl\_odeiv\_step\_rk4}, the 4th order (classical)
Runga-Kutta methods. 

For each timestep, we evolve the the three equations first. And the run
the network to get the entropy generation and abundance changes. We use
both $x$, $y$, $s$ and species abundances to control the timestep. 

Figure \ref{fig:x} shows how the size of the star changes with time. This
calculation and the following few are all with an initial density of
$9\times 10^9$g/cm. In this figure the normalized radius $x$ starts to increase
around $10^{-2}$ seconds and become $10^8$ times of original in about one
year ($~3\times 10^8$ seconds). 

\begin{figure}[h!]
\centering
\includegraphics[width=\figuresize\textwidth]{figures/x.pdf}
\caption{
The normalized radius change with time for a calculation of the simple type Ia
model. 
}
\label{fig:x}
\end{figure}

Figure \ref{fig:v} shows the change of velocity with time. The expanding 
speed of the star shoots up at about $10^{-5}$ seconds after the calculation
starts and becomes significantly fast (~$10^8$cm/s) around $10^{-2}$ seconds.
Due to the weak interaction (as shown below) the velocity varies a little bit
and peaks around 1 second, and then levels at about $3\times 10^8$ cm/s,
~1% of speed of light. 

\begin{figure}[h!]
\centering
\includegraphics[width=\figuresize\textwidth]{figures/v.pdf}
\caption{
The velocity change with time for a calculation of the simple type Ia
model. 
}
\label{fig:v}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[width=\figuresize\textwidth]{figures/T.pdf}
\caption{
The temperature change with time for a calculation of the simple type Ia
model. 
}
\label{fig:T}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[width=\figuresize\textwidth]{figures/T3.pdf}
\caption{
The temperature change with time for a calculation of the simple type Ia
model for various initial densities. 
}
\label{fig:T3}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[width=\figuresize\textwidth]{figures/Trho3.pdf}
\caption{
The temperature change with density for a calculation of the simple type Ia
model for various initial densities. 
}
\label{fig:Trho3}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[width=\figuresize\textwidth]{figures/mass_rho.pdf}
\caption{
The mass fraction change with initial densities for calculations of the 
simple type Ia model.
}
\label{fig:mass_rho}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[width=\figuresize\textwidth]{figures/radioactive_mass.pdf}
\caption{
The mass fraction of radioactive nuclei vs. initial densities for 
calculations of the simple type Ia model.
}
\label{fig:radioactive_mass}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[width=\figuresize\textwidth]{figures/yield.pdf}
\caption{
The mass fraction of selected isotopes vs. mass coordinates for a polytropic
profile of white dwarf.
}
\label{fig:yield}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[width=\figuresize\textwidth]{figures/yield_half.pdf}
\caption{
The mass fraction of selected isotopes vs. mass coordinates for a polytropic
profile of white dwarf, with slower expansion (half of the speed).
}
\label{fig:yield_half}
\end{figure}

